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Consider the Heston model and the Local Volatility model with local volatility built (using Dupire) from the Heston reconstructed vanilla options implied volatility. The price of any European payoff will be the same under both models. The price of exotic options will usually not be the same.


You're implementing the formula $$C=S_0e^{-qT}\Pi_1-Ke^{-rT}\Pi_2$$ where \begin{align*} \Pi_1&=\frac{1}{2}+\frac{1}{\pi}\int_0^\infty \text{Re}\left(\frac{e^{-iu\ln(K)}\varphi_{\ln(S_T)}(u-i)}{iu\varphi_{\ln(S_T)}(-i)}\right)\text{d}u,\\ \Pi_2&=\frac{1}{2}+\frac{1}{\pi}\int_0^\infty \text{Re}\left(\frac{e^{-iu\ln(K)}\varphi_{\ln(S_T)}(u)}{iu}\right)\...

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